Calculus Examples

$f (x) = x {(x - 15)}^{2}$

Step 1

Rewrite ${(x - 15)}^{2}$ as $(x - 15) (x - 15)$ .

$\frac{d}{d x} [x ((x - 15) (x - 15))]$

Step 2

Expand $(x - 15) (x - 15)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [x (x (x - 15) - 15 (x - 15))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [x (x \cdot x + x \cdot - 15 - 15 (x - 15))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [x (x \cdot x + x \cdot - 15 - 15 x - 15 \cdot - 15)]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [x (x^{2} + x \cdot - 15 - 15 x - 15 \cdot - 15)]$

Step 3.1.2

Move $- 15$ to the left of $x$ .

$\frac{d}{d x} [x (x^{2} - 15 \cdot x - 15 x - 15 \cdot - 15)]$

Step 3.1.3

Multiply $- 15$ by $- 15$ .

$\frac{d}{d x} [x (x^{2} - 15 x - 15 x + 225)]$

Step 3.2

Subtract $15 x$ from $- 15 x$ .

$\frac{d}{d x} [x (x^{2} - 30 x + 225)]$

Step 4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x^{2} - 30 x + 225$ .

$x \frac{d}{d x} [x^{2} - 30 x + 225] + (x^{2} - 30 x + 225) \frac{d}{d x} [x]$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $x^{2} - 30 x + 225$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [225]$ .

$x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [225]) + (x^{2} - 30 x + 225) \frac{d}{d x} [x]$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x (2 x + \frac{d}{d x} [- 30 x] + \frac{d}{d x} [225]) + (x^{2} - 30 x + 225) \frac{d}{d x} [x]$

Step 5.3

Since $- 30$ is constant with respect to $x$ , the derivative of $- 30 x$ with respect to $x$ is $- 30 \frac{d}{d x} [x]$ .

$x (2 x - 30 \frac{d}{d x} [x] + \frac{d}{d x} [225]) + (x^{2} - 30 x + 225) \frac{d}{d x} [x]$

Step 5.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x (2 x - 30 \cdot 1 + \frac{d}{d x} [225]) + (x^{2} - 30 x + 225) \frac{d}{d x} [x]$

Step 5.5

Multiply $- 30$ by $1$ .

$x (2 x - 30 + \frac{d}{d x} [225]) + (x^{2} - 30 x + 225) \frac{d}{d x} [x]$

Step 5.6

Since $225$ is constant with respect to $x$ , the derivative of $225$ with respect to $x$ is $0$ .

$x (2 x - 30 + 0) + (x^{2} - 30 x + 225) \frac{d}{d x} [x]$

Step 5.7

Add $2 x - 30$ and $0$ .

$x (2 x - 30) + (x^{2} - 30 x + 225) \frac{d}{d x} [x]$

Step 5.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x (2 x - 30) + (x^{2} - 30 x + 225) \cdot 1$

Step 5.9

Multiply $x^{2} - 30 x + 225$ by $1$ .

$x (2 x - 30) + x^{2} - 30 x + 225$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$x (2 x) + x \cdot - 30 + x^{2} - 30 x + 225$

Step 6.2

Combine terms.

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Step 6.2.1

Raise $x$ to the power of $1$ .

$2 (x^{1} x) + x \cdot - 30 + x^{2} - 30 x + 225$

Step 6.2.2

Raise $x$ to the power of $1$ .

$2 (x^{1} x^{1}) + x \cdot - 30 + x^{2} - 30 x + 225$

Step 6.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{1 + 1} + x \cdot - 30 + x^{2} - 30 x + 225$

Step 6.2.4

Add $1$ and $1$ .

$2 x^{2} + x \cdot - 30 + x^{2} - 30 x + 225$

Step 6.2.5

Move $- 30$ to the left of $x$ .

$2 x^{2} - 30 \cdot x + x^{2} - 30 x + 225$

Step 6.2.6

Add $2 x^{2}$ and $x^{2}$ .

$3 x^{2} - 30 x - 30 x + 225$

Step 6.2.7

Subtract $30 x$ from $- 30 x$ .

$3 x^{2} - 60 x + 225$